Wednesday, June 17, 2009

The Unreasonable Effectiveness of the Human Brain

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."

Though he was indeed speaking about two miracles

"[...] the two miracles of the existence of laws of nature and of the human mind's capacity to divine them."

And while a lot has been said and written about the fact that the laws of nature seem to be formulated in the language of mathematics, I always found the larger miracle to be that we simple humans are able to grasp these laws, to write them down, and to use this knowledge to shape Nature according to our liking. As far as mathemetics is concerned, if it wasn't unreasonably effective to that end would we be wondering about it? If mathematics wasn't good to describe the real world it would just be a bizarre form of art with a nerdy cult. The actual question is why is there anything so unreasonably effective that the human brain can comprehend. Understanding, say, cosmological perturbation theory or the representations of Lie Groups isn't such a large survival advantage. (And then there's those who claim we just do it all for the sake of reproduction, obviously.)

Let me highlight two extreme points of view one can take on our ability to describe Nature. Max Tegmark's hypothesis of the "Mathematical Universe" puts forward the idea that mathematics is really all there is, claiming that mathematics is the only thing that is ultimately "free of human baggage". And if we find that counterintuitive or disturbing, that's because our brains didn't evolve to understand the fundamental nature of reality. In fact, according to Tegmark, we should be disturbed if we did not find the fundamental description of reality puzzling.

I wrote previously that though the hypothesis can of course stand as such, the justification Tegmark gives is either empty or logically faulty. If one defines what is free of human baggage to be mathematics, then it's an empty statement, thus let's not go there. Without that, we don't know - we can't know - whether mathematics is the only language free of human baggage. It remains possible there exists an even more fundamental language that relates to today's math in a way that today's math relates to narrative. And no, I can't tell you what that would be. But then, our brains didn't evolve to understand...

I actually think Tegmark isn't quite consistent on that point. One the one hand he is defending his hypothesis by saying fundamental reality should seem bizarre to our human brains that evolved to hunt bears, but then the idea that there's something more fundamental than math is too bizarre for him. Do you think 50000 years ago when humans had just begun to use spoken language and to scratch pictures into stones, many of them would have been sympathetic to the idea there's something more powerful than these ingenious achievements of the human mind that allowed them to communicate relations between things without actually pointing at them, and even talk about things that they might have entirely invented?

Now let's look at it from the completely other perspective. A lot of theoretical physicists like to talk about "naturalness," "elegance" or "beauty" of a theory or an equation. These are arguably human judgements and often perceptions that are considerd very important. It is also interesting that, given the required educational background, most people tend to agree on these terms (modulo a deliberate or opportunistic self-deception that financial or peer pressure can create). Whether it is reasonable or not, they do implicitly assume that the human brain has a sense about Nature's ways and that their intuitions will lead them a way to success.

I'm not sure why that would be except possibly that after all our brains are made of the same stuff as elementary matter and we are part of the universe we aim to describe and in constant interaction with it. What we are trying to do is to create an accurate image of the universe in our brains, an imperfect repetition of a structure within a structure of that system. Then the question we arrive at is why does the universe evolve to create subsystems that to increasing accuracy mirror the properties of the system? (And, can you continue this self-similarity both up- and downwards?)

Well, needless to say, I can't give you an answer for why the human brain is so unreasonably effective in understanding the laws of Nature. And indeed, spending most of my time between people who believe they have the key to understanding the universe, I sometimes wonder whether our ambitions will continue to deliver insights or whether we'll eventually reach the limits of what we can comprehend, endlessly fooling ourselves into believing we're getting closer to unraveling the fundamentals. But in any case, we're far from reaching that limit. Give it another 50000 years or so.

Somehow it doesn't seem reasonable that the differential equations that represent the laws of physics should contain as one of their 'emergent solutions' self-assembling blobs of animate matter (us) that are able to deduce and write down those very equations. And yet this is apparently what occurs. The existence of self-referential solutions is, to me, the real miracle.

Uterine development mirrors evolution (ontogeny recapitulates philogeny) to fabricate a final compromise. The human brain within is a messy jury rig of successively more sophisticated structures. If you want different if not better understanding you must build an efficient brain without the old baggage.

At least 10% of the First World can make no sense of mathematics at all. We goad their reproduction through Head Start. Rather than foster brilliance we allocate for its suppression. Both have been unreasonably effective as the world abandons engineering solutions for social advocacies and tests of faith.

Here's the thing. As one explores the theoretical world from my perspective such abstraction seem to be the "basis of experience" and movement toward real world dynamical revelation according to the experiments one ventures froward to set up.

Stephen Wolfram argues that the way to unlock the rest of science is to give up on mathematics and look for explanations analogous to computer code. Very simple computer programs can produce remarkably complex behaviour that mimics phenomena science has had difficulty modelling, like the motion of fluids, for example. So studying the behaviour of these programs may provide scientists with new insights about these phenomena. Indeed, Wolfram thinks the universe itself may be generated by a computer program simple enough to be expressed in a few lines of code. “If the laws are simple enough, if we look in the right way we’ll find them,” he says. “If they’re not, it will be tougher. The history of physics makes one pessimistic that we could ever end physics. I don’t share that pessimism.”

But before this, the underlying fundamental "basis of reality" is questioned as to it's substance? Frank Wilczek comes to mind here on his Theory of Matter. Ya I know, Bla bla:)

So early on, I seen this relation as to the need for experimental validation arising out of super strings theory's quest for some kind of validation. Witten too, sought for such correspondence in relation to the "condense matter theorist view" and so did Howard Burton's choice, on the title for his new book First Principles. You see?

Because Lee isolates such desire in your "this and that articles of choice," toward the mathematical mind," where is the true source of the effectiveness if mind cannot take a position? Then, push perspective forward while still remaining responsible and effective to dealing with the real world. Never mind the "first three seconds of Steven Weinberg" but now of the "three microseconds" and you see the effectiveness of pushing back perspective in a reductionistic mode.

The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner

The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

[3 M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: "All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting" (p 188).]

That laws of nature exist is clear even to primitive humans 100,000 years ago. It is easy to see everything operate in a 'consistent' way. The stars in the sky are always there.

But human development of mathematics to comprehend nature is a stroke of luck. While the human minds have been capable of inventing simple mathematics for many centuries (witness Egyptian pyramid construction 5000 years ago), it has failed to make any significant headways into beyond cursory understanding of nature until Newton's time. This is a bleep in time in the history of human. And even after Newton, much of the world great minds cannot (or refused to) understand his work. Religious dogma played a great role in preventing mathematics from moving forward.

It was only in the 20th century that human finally discovered the true power of the math invented. It is a very lucky thing.

But is nature purely mathematical. If one is talking *our* math, of course not. Other advanced civilizations, with a different mental design and construction, will invent *their* math and will discover the laws of nature in a different way. Our understanding of quantum phenomenon is described (and thus limited) by the wave probability function - our math. It would be arrogant to think this is the only possible math. The fact we haven't yet figure out a different, more complete or accurate description does not mean it does not exist. Our math has equations. Who way this is the only possible math?

Now a Platonist would say that chess always existed timelessly in an infinite space of mathematically describable games. We do not achieve anything by believing that, except an emotion of doing something elevated.The unique universe by Lee Smolin

I don't know, but I kind of think this is a cheap shot at what theoreticians can do to reduce something too, to a relativistic perspective , and what geometrician like Coxeter thought about polytopes and such?:) So Garrett likes to see in "abstract spaces" and compare Kaleidoscopes while dreamingly making expensive surf boards?;)

Mathematics can describe only deterministic things completely and what we know about quantum mechanics world is, it's indeterminism is fundamental and experimentally proven to agree with Bell's theorem. In addition, by Gödel's theorems even the formal math as such has it's own limits, as it cannot describe itself completely. By AWT such similarity isn't accidental, because number concept is based on countable objects, i.e. concept of colliding particles fulfilling Fermi-Dirac statistics. Formal math has nothing very much to say about bosons, which are violating Peano algebra like ripples at water surface: they cannot be counted. In my opinion formal math isn't equivalent, but subgroup of physical reality, which is subgroup of observable world. Not all things which we can feel we can observe, not all things, which can be observed (a physical reality) can measured, not all things which we can measure can be measured in reproducible way, not all things, which we can measure can be computed, not all things, which can be computed can be falsified. In addition, not all things, which can be computed numerically can be derived in formal way (for example the formal model of N gravitational bodies). Formal math is in fact quite poor subclass of observable reality - which shouldn't prohibit us in extending of its relevance scope, whenever possible.

By AWT human brain is based on electromechanical solitons (sound wave packets), analogous to quantum wave packets of particles in vacuum. In certain sense, brain is an analog simulator of gradient driven reality in small scale, which enables to interact with it quite effectivelly, as it comprehends both transversal, both longitudinal wave spreading model of information (i.e. both consecutive logics, both parallelistic intuition).

1. I am very unconvinced by the MUH/modal realism perspective. First, our universe really can't be well described/modeled by math anyway, believe it or not.

a. Math structures cannot produce true randomness. For example, I can't create a "mathematical structure" that will produce different results each time it is "run" or applied. I can create a string of "pseudorandom" digits by taking sqrt 23 etc. but each time I take sqrt 23 it's the same sequence! Well of course, since math is based on logical necessity. An identical "entity" in math must give an identical result. But real objects like muons in our universe, can be identical and yet behave differently. One decays after one duration, another lasts a different duration. What, are there different "seed" generators inside each muon?

(Readers, don't be confused by there being math to deal with statistics etc. Those maths either give the proportions of probable outcomes rather than produce actual unpredictable outcomes, or they deal with the summed overall result of combined instances of true randomness. Nor are means of “cheating” like simply declaring a fiat “random variable” a way of actually generating the intrinsically and *in principle* unpredictable outcomes as in our real universe!

b. The wave function is not rational, since we can’t describe its distribution in space-time throughout the entire process that includes a measurement. Sure, a Schrödinger equation describes the evolution of a WF up until it gets “measured”, but then what happens to the WF? Would a computer or “Platonic” simulation just make the WF vanish or suddenly change (in whose reference frame ....), how would it deal with that? Note that the pretense of decoherence being a solution just takes the similar given outcomes from scrambled phases and “classical systems” to produce a deceptive circular argument. It doesn’t say what happens to those extended WFs as a function of time or explain why they do get localized. So what if they are not in a well-ordered coherent superposition, the wave components are still *extended* and localizing them into a little spot is the problem, not their relative phases. I challenge any decoherence enthusiast to produce a simulation showing all the wave amplitudes in time and space throughout the process, including the measurement. How will you “stuff” the WF into a small space or change it from large and strict momentum to small and spread momentum, etc? Even if you say one component runs off into "another world", can you model it continuously?

c. More a matter of opinion: the nature of conscious experience, qualia etc., cannot be modeled as “math”. There is no computation which corresponds to “nausea” or the experienced quality of green etc. Some, like the IMHO shallow denialist Daniel Dennett say there is no difference, but most of us realize e.g. that there’s something worth being afraid of or worth seeking with feelings, that just having “information” cannot correspond to. The issue of our minds leads to:

...2. Our minds can comprehend the universe because of the latter’s orderly nature – for the most part! But to me, what is most fascinating is our ability to be sure “we really exist”, that we aren’t just “conceptual entities as in MUH. Note that if our minds were simply AI computational entities, there'd be no way for us to even have the thought that we *weren’t* just math. However incredible, it is a rigorous necessity that computations and logic don’t have a representation for “realness” in the “material” sense. Ultimately, numbers (more broadly, set theoretic entities) are just being worked on. A computation or math representation can only correspond to a mathematical fact. It wouldn’t matter whether it’s a real “me” shooting a “real” robot in a material universe, or a representational me shooting a robot in a “game” simulation – the same representation has to be used.

Hence, this is like the issue of “zombies” in the philosophy of mind. A “zombie” means a being which acts just like us, and has no discernible distinctions internally, but is “not really conscious.” IOW, it doesn’t really have feelings, experience. It is a conceptual tool apart from whether the same real universe can contain both types of being. It has been argued that if we are really just AI mentalities, we cannot even understand or represent the difference, and it is an “illusion.” But if we are AI, then neither could we really have a coherent thought or represent that our universe is “real” in some sense beyond MUH and modal realism. IOW, a math-mind cannot have a thought (computation) that legitimately expresses the distinction between MUH and the old intuition that “material existence” is something special beyond mere conceptual description. So a “materially real” universe is akin to a “zombie” in PoM – it can’t be mathematically or logically distinguished from a math-universe (mathiverse?) with the same “features” (configuration of quantities etc.)

You may be thinking, “no way”, but I challenge anyone to make a strictly logical distinction, that is not a circular argument, of how we formally define the difference between an incarnate “material universe” and a Platonic mathiverse. But of course, they are only logically equivalent if the universe *can* be described formally. I gave reasons above to believe it can’t be. Maybe that ties our sense of “real conscious experience” (that we are not zombies) in with our sense that MUH/modal realism is wrong – that our world is more real that conceptual worlds that could just as easily be mere programs running in a computer.

Mathematics (ie. platonic metaphysical super-reality) has been formally shown to be infinitely complex. How can infinite complexity arise from a finite material universe? I've always thought this as an indication that the material universe (as described by physics) is a subset of mathematics.

Why are quantitative measures of nature related by quantitative laws that are universally applicable? - this trivializes the problem and misses some essential aspect of it.

Another possibility - we humans aren't very good at mathematics, and out of the vast sea of math have only uncovered stuff that is relevant to nature. Maybe if we were good at mathematics, our question would be - why is so little of mathematics applicable to describing nature?

Why do humans seem to have abilities not directly tied to fitness in the evolutionary sense? Or perhaps, "what ability is it that gives a survival edge to humans, but, as a side effect, also enables humans to do mathematics?"

Are there any universally applicable rules of nature that are not mathematical in nature? ("Beware of large carnivores!" :) might be one such.)

In what circumstances might we be pondering the curiously moderate effective of mathematics in describing nature?

Only questions, no answers. I may have given away what my inclinations are.

First, the math structure does not "arise" from our universe! It is a logically necessary, eternal entity that's "just there". The problem is, is our universe merely part of that giant superstructure, or is it beyond math? I gave arguments (like true randomness) that indicate our universe is more than just math/describable as math. Our universe can't be a subset of math since math doesn't produce variable results from the same starting state, etc.

Janne, infinite complexity is not the same as "randomness". A number like sqrt two has infinite complexity since the digits go on forever, and there are Aleph null of them: 1.4142135623730950488016887242097...But it's always the same sequence of digits, every time you take the sqrt of two! In real randomness, you cannot know in principle what will happen, it is "not in the cards" already stacked up like the digits of sqrt 2, pi, etc.

BTW readers can find about Borel's find via www.arxiv.org/pdf/math/0506552. It isn't quite the same as my point, but my argument stands on its own.

A nice piece, where you basically explore what for many appear as opposing viewpoints. The first being the world is essentially a mathematical construct and the second that it being what you called “natural” which has been referred to by Plato of antiquity as the aspects of truth and beauty or what Robert Pirsig more recently refined and compressed to be “quality”. Despite them looking as polar opposites what they share in common is reason, which we can also call logic.

That is mathematics describes the world so well as it’s based on logic and truth and beauty or quality if you prefer are the unavoidable products of logic. For me then the hardest question to face is one that you actually dance around a bit here and that is to ask why is it that reality must be logical; which for me is a greater mystery than to ask why we are capable to recognize it as being such. That is you yourself already suggested the answer, for that being since we are simply one of many of its logical products, (consequences) only in this case one that is able to think.

I then find no mystery in there being thinking creatures to exist within a reality manifest of logic. As a matter of fact it would seem to me to be a logical mandatory requirement or eventuality you might say. To frame the question in the new quantum context, is to ask if all is information, what or who is it to inform? The question then for science to ask as to explore is if a reality or realities can exist that is not manifest of logic?

Yes that's a good pointer, Chaitin's work deals in depth with these issues.

But sqrt(2) isn't infinitely complex, even though the digits go on forever. The argument goes that you can produce finite amount of information (for example a computer program) that can reproduce all the digits, thus contain all the information of that real number. You can do the same with pi for example. You can't necessary get all the digits in practice (with finite time etc) but it doesn't matter, the information of the digits is still contained with finite amount of information.

These numbers Borel showed can't be contained like this, so they contain infinite amount of information. So there is true randomness to the digits, this is how I seem to get it.

Janne, you're right, Borel's numbers aren't the same as just the digits of roots, pi, etc. But they still don't show *variation* from one time to another. The number "is what it is" - you can manipulate it and it may contain infinite information, but the number itself it is the same always. Don't confuse infinity or complexity with the sort of randomness, that the same initial state can produce different results in different instances.

BTW, it is true that the number of digits of irrational #s is indeed Aleph null - it is the set of irrational numbers themselves, that cannot be of cardinality Aleph null. Anyone have preference, over whether continuum is Aleph one or not? Heh, we can't seem to be sure ...

In the Bohmian perspective random is just another name for the set of all orders and if accepted would encompass all that would or could be consider as reason or reasonable. The question then to ask if a set can exist that is beyond this as not being reasonably definable with infinity as the limit and random as the set? I would say no to that, and yet not having any way to prove it as such.

There is no discussion also of what mathematics is being compared against in being unreasonably effective. (The only other way of making sense of the question is why is mathematics not only moderately effective in describing nature?)

I suppose the answer is - compared to revelation (the holy books) and compared to philosophy?

That’s precisely the point, being as it could be true that random may not be quantifiable and therefore beyond the bounds of mathematics, yet still is able to be qualified. As for example, a random sequence as being a true subset of random is one not able to be defined by elements of information (explanation) less in number then itself.

Therefore, on one hand it may be reasonable (logically consisitant) to insist it’s beyond the scope of mathematics, yet another to claim this extends to it being beyond reason itself. So if it still is bounded by reason, yet not predictable within number, as able to predict, does this necessitate it as not to be determined? Further, when reason is involved, I also think that predetermined can be seen to be different then determined, as the first suggests being without choice, while the latter may require it.

I find this a depressing subject, for the following reason. Let's assume that there is no fundamental relationship between the basic laws of nature and the capacities of the human brain. It follows that those fundamental laws should be easy to understand, or impossibly difficult --- it would be a ridiculous coincidence if, say, only the top 1000 brains on earth could understand them. Now take one look at the arXiv to see which of the two possibilities is correct.........

So early on, I seen this relation as to the need for experimental validation arising out of super strings theory's quest for some kind of validation. Witten too, sought for such correspondence in relation to the "condense matter theorist view" and so did Howard Burton's choice, on the title for his new book First Principles. You see?

That's the thing, while the abstractions are trying to explain the natural process, and you were gifted with "insight" into this new natural conception, how is it that such a thing could have enter from an abstract process? You had to arrive at a certain point and then, it comes to you.

Well, I think Wolfram partly disagrees with himself, and where he does it's where Lee's article becomes important. See, exactly because the phenomena emerging from a simple set of rules can be very complex, it might be impossible to reduce these phenomena to the simple rules. In fact, it might be impossible to ever deduce a 'meta law' for the evolution of laws from observations that we can possibly make.

Now I've written earlier on Emergence and Reductionism, where I've pointed out that even if there is an evolution law, it might be impossible to determine it, which in practice amounts to saying we have no law. Best,

These are interesting questions. I think that a lot of discussion stems from the fact that 'mathematics' itself isn't well-defined. Besides that, being sexy might be necessary but isn't sufficient for successful reproduction. Best,

This has indeed been a touching subject with regards to positions adopted for moving forward.

As I mentioned earlier in regards to Veneziano, and the idea of something that comes "before this universe" this is indeed a touchy subject for cosmologists, yet, it should not be such a problem when identifying a movement toward the limits of reductionism as an understanding about that Quark soup as Phil mentions there in the reference link Bee.

You see I looked for where that extra energy can go, if we are thinking in terms not only of the limits of reductionism but of what superconductors of themself imply in relation to the quark soup there had to be commonality between them. So I call it a place where things come into this universe. You see?

Memories arise out of an equilibrium?:) Was a philosophical perspective about what can transfer from one universe to another. You see, all this information has to exist somewhere? Tegmark might call it a museum, while I might call it a Library?:)

Bee said "... If mathematics wasn't good to describe the real world it would just be a bizarre form of art with a nerdy cult. ...".

As to the "form of art" that is true mathemantics, see the paper "A Mathematician's Lament" by Paul Lockhart at

http://plato.asu.edu/LockhartsLament.pdf

is from the point of view of a mathematician who considers math to be a "form of art" like music and painting, and compares how utilitarian math education (prevalent in the USA and some other countries) suppresses the true art of math.

Tony, yea maybe with a little Ars Magna in math education, somethimg like what I wrote earlier today on an Enneagram board might seem a little more normal for a wider audience:

If you think you think air - water - other for head - heart - gut and use fire - earth for N - S and Fixed - Mutable for J - P then for Pisces through Aquarius you get:

331749952866

Looking at the Trines this gives:

342398196756

A heart triad, a head triad, and some strange two trine version of a gut triad. I like the idea various people have for two laws of 3 but I still prefer one 891 gut triad. I think these two "gut trines" might be related to the Enneagram idea of the 3 and 6 being swappable.

Math as art, an interesting way to imagine it. Then again there is a difference between the expression of art and what it’s expressing. Also, with art one has the media to consider, with music it’s action onto air, with poetry words onto paper, while with painting pigment onto canvas with the only commonality being from where they all manifest. So what then is the media of mathematics other then thoughts onto mind. In this respect should we not consider mathematics as being meta-art rather than art? Then again perhaps we should make the distinction between a work of art and art itself.

Just as an afterthought formed of recent experience, which relates to what Paul Lockhart has lamented. In as I work in manufacturing, there are several engineers and draftpersons that are employed with us, of which I’m not one of. Over the course of my dealings with them on a regular basis I was brought to suspect that although they incorporate mathematics often in what they do, they seemed not to understand what it was from a fundamental perspective. Being curious if my suspicion had any merit I asked several of them a question which went like this.

I explained the task is to form what would be to a good approximation a right angle, with all that they are given being a length of string, with no other devises or instruments permitted to be used other than if required the physical assistance of someone else. I must sorely report that not one of them offered an answer to my query, yet worse a few appeared somewhat puzzled even after being given the solution. It seems that with the advent of aids such as AutoCad that even the most rudimentary of geometric relationships resides now only in programs they use. So Lockhart’s point is well taken, yet I see this as not only an indictment, yet sadly the sentence for much of humanity.

Course it's probably the other Ars Magna, but no law against teaching both. I once taught Tony Smith's physics in a religion class, the only day the kids looked genuinely interested (also the only day I looked genuinely interested).

If you count grains of sand on a beach, is the emerging mathematical structure "unreasonably effective"? Or did you make it effective with approximations such as ignoring differences between grains or the unknown-but-large upper bound to their number?

This is the big flaw in Tegmark's argument. His "baggage free" reasoning would hold if laws of physics didn't all depend on baggage such as: what is a kilogram, what is a field, what is a grain of sand. Unfortunately, they do.